Why a Sphere?
Every single-qubit pure state can be written as:
|ψ⟩ = cos(θ/2)|0⟩ + eⁱᵠ sin(θ/2)|1⟩
Two real parameters — θ (polar angle, 0 to π) and φ (azimuthal angle, 0 to 2π) — fully describe any qubit state. Two parameters define a point on the surface of a sphere. That's the Bloch sphere.
The Bloch sphere is a unit sphere in 3D space. Every point on its surface is a valid pure qubit state. Points inside the sphere represent mixed (partially decohered) states — this becomes relevant when a qubit is entangled with others.
The Three Axes
The Bloch sphere axes correspond to the three Pauli operators:
| Axis | Poles | Gate that rotates around it | |------|-------|--------------------------| | Z (vertical) | North = |0⟩, South = |1⟩ | Rz(θ), Z gate | | X (horizontal) | +X = |+⟩, −X = |−⟩ | Rx(θ), X gate | | Y | +Y = |i⟩, −Y = |−i⟩ | Ry(θ), Y gate |
The Z-axis is special: it's the measurement axis. Measuring in the Z basis collapses the qubit to north (|0⟩) or south (|1⟩).
Key States on the Sphere
- North pole (z=1): |0⟩ — the ground state, all probability in |0⟩
- South pole (z=−1): |1⟩ — the excited state
- +X equator (x=1): |+⟩ = (|0⟩+|1⟩)/√2 — equal superposition, positive phase
- −X equator (x=−1): |−⟩ = (|0⟩−|1⟩)/√2 — equal superposition, negative phase
- +Y equator (y=1): |i⟩ = (|0⟩+i|1⟩)/√2 — S gate applied to |+⟩
Gates as Rotations
Every quantum gate is a rotation of the Bloch vector:
The Hadamard gate rotates the vector by 180° around the axis midway between X and Z — turning |0⟩ into |+⟩ and |+⟩ back into |0⟩.
The Pauli-X gate (quantum NOT) flips the vector from north to south — taking |0⟩ to |1⟩.
Rotation gates (Rx, Ry, Rz) let you apply a precise angle of rotation around each axis.
Purity and Entanglement
For a single qubit in isolation, its Bloch vector always has length 1 — it sits on the sphere surface. But when you have multiple qubits that are entangled, looking at one qubit alone gives a reduced state whose Bloch vector is shorter than 1.
A Bloch vector of length 0 (the center of the sphere) means the qubit is maximally entangled — you have zero information about its state without knowing the other qubit.
Purity indicator: The purity bars in the Bloch Sphere Simulator show the length of each qubit's Bloch vector. Apply a CNOT gate (with 2 qubits) after putting qubit 0 in |+⟩ — watch the purity drop as they entangle.
Quiz
A qubit's Bloch vector points in the +X direction. What state is it in?
Try It
Open the Bloch sphere simulator and verify these claims yourself:
Press H to apply Hadamard — watch |0⟩ (north pole) move to the +X equator. Press X to apply Pauli-X — the vector flips to the south pole.